cr li-yau gradient estimate and perelman entropy …ohnita//2011/slides/part1/shu...cr li-yau...

CR Li-Yau Gradient Estimate and Perelman EntropyFormulae

Shu-Cheng Chang

National Taiwan University

The 10th Paci�c Rim Geometry ConferenceOsaka-Fukuoka, Part I, Dec. 1-5, 2011

Shu-Cheng Chang (National Taiwan University)CR Li-Yau Gradient Estimate and Perelman Entropy FormulaeThe 10th Paci�c Rim Geometry Conference Osaka-Fukuoka, Part I, Dec. 1-5, 2011 1

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Contents

MotivationsPseudohermitian 3-ManifoldThe CR Li-Yau Gradient EstimateThe CR Li-Yau-Hamilton and Li-Yau-Perelman Harnack EstimatePerelman Entropy Formulas and Li-Yau-Perelman Reduce DistanceThe Proofs


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Motivations

Problemgeometrization problem of contact 3-manifolds via CR curvature �ows

The Cartan Flow : Spherical CR structure

The torsion �ow : the CR analogue of the Ricci �ow


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The Torsion Flow

The torsion �ow �∂tJ(t) = 2AJ ,θ∂tθ(t) = �2W θ(t)

.

Here J = iθ1 Z1 � iθ1 Z1 and AJ ,θ = A11θ1 Z1 + A11θ1 Z1.


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The CR Yamabe Flow

In particular, we start from the initial data with vanishing torsion :�∂tJ(t) = 0∂tθ(t) = �2W θ(t)

.

The CR Yamabe Flow (Chang-Chiu-Wu, 2010, Chang-Kuo, 2011)

∂tθ(t) = �2W θ(t).


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Poincare Conjecture and Thurston GeometrizationConjecture via Ricci Flow

Sphere and Torus decomposition

Singularity formation1 Li-Yau gradient estimate for heat equation ( 1986)2 Hamilton-Ivy curvature pinching estimate (1982, 1995)3 Hamilton Harnack inequality ( 1982, 1988, 1993, etc)4 Perelman entropy formulae and reduce distance (2002, 2003)

Geometric surgery by Hamilton and Perelman


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Geometrization problem of contact 3-manifolds

Contact Decomposition theorem and Classi�cation

CR Geometric and Analytic aspects :1 Existence of a " best possible geometric CR structure" on closedcontact 3-manifolds- spherical CR structure with vanishing torsion.

2

Rij : Ricci curvature tensor $ A11 : pseudohermitian torsion

ProblemSub-Laplacian ∆b is degenerated along the missing dirction T bycomparing the Riemannian Laplacian ∆.


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geometrization problem of contact 3-manifolds

ProblemWe proposed to deform any fixed CR structure under the torsion on a contactthree dimensional space which shall break up due to the contact topologicaldecomposition.

ProblemThe asymptoic state of the torsion flow is expected to be broken up into pieceswhich satisfy the spherical CR structure with vanishing torsion.

ProblemThe deformation will encounter singularities. The major question is to find away to describe all possible singularities.


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Pseudohermitian 3-manifold

Let (M, J, θ) be the pseudohermitian 3-manifold.1 (M, θ) is a contact 3-manifold with θ ^ dθ 6= 0. ξ = ker θ is called the

contact structure onM.2 A CR-structure compatible with ξ is a smooth endomorphismJ : ξ ! ξ such that J2 = �identity .

3 The CR structure J can extend to C ξ and decomposes C ξ intothe direct sum of T1,0 and T0,1 which are eigenspaces of J withrespect to i and �i , respectively.


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Given a pseudohermitian structure (J, θ) :1 The Levi form h , iLθ

is the Hermitian form on T1,0 de�ned by

hZ ,W iLθ= �i

dθ,Z ^W

�.

2 The characteristic vector �eld of θ is the unique vector �eld T suchthat θ(T ) = 1 and LT θ = 0 or dθ(T , �) = 0.

3 Then fT ,Z1,Z1g is the frame �eld for TM and fθ, θ1, θ1g is thecoframe.


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The pseudohermitian connection of (J, θ) is the connection rψ.h.

on TMC (and extended to tensors) given by

rψ.h.Z1 = ω11Z1,rψ.h.Z1 = ω1

1Z1,rψ.h.T = 0

with

dθ1 = θ1^ω11 + A11θ^θ1

ω11 +ω1

1 = 0.

Di¤erentiating ω11 gives

dω11 = W θ1^θ1 (mod θ)

where W is the Tanaka-Webster curvature.


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We can de�ne the covariant differentiations with respect to thepseudohermitian connection.

1

f,1 = Z1f ; f11 = Z1Z1f �ω11(Z1)Z1f .

2 We de�ne the subgradient operator rb and the sublaplacianoperator ∆b

rb f = f,1Z1 + f,1Z1,and

∆b f = f,11 + f,11.


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ExampleD is the strictly pseudoconvex domain

D � C2 and M = ∂D

withD = fr < 0g and M = fr = 0g.

Chooseξ = TM \ JC2TM and θ = �i∂r jM

withJ = JC2 jξ .


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Li-Yau Harnack Estimate

Theorem(Li-Yau, 1986) The Li-Yau Harnack estimate

∂(ln u)∂t

� jr ln uj2 + m2t� 0

for the positive solution u(x , t) of the time-independent heat equation

∂u (x , t)∂t

= ∆u (x , t)

in a complete Riemannian m-manifold with nonnegative Ricci curvature.


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Li-Yau-Hamilton Inequality

Theorem( Hamilton, 1993) Hamilton Harnack estimate (trace version)

∂R∂t+Rt+ 2rR � V + 2Ric(V ,V ) � 0

for the Ricci �ow∂gij∂t

= �2Rij

on Riemannian manifolds with positive curvature operator.


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Subelliptic Li-Yau gradient estimate

Consider the heat equation

(L� ∂

∂t)u (x , t) = 0

in a closed m-manifold with a positive measure and an operator withrespect to the sum of squares of vector �elds

L =l

∑i=1X 2i , l � m,

where X 1, X2, ...,Xl are smooth vector �elds which satisfy Hörmander�scondition : the vector �elds together with their commutators up to �niteorder span the tangent space at every point of M.


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Subelliptic Li-Yau gradient estimate

Theorem(Cao-Yau, 1994) Suppose that [Xi , [Xj ,Xk ]] can be expressed as linearcombinations of X 1, X2, ...,Xl and their brackets [X1,X2], ..., [Xl�1,Xl ].Then, for the positive solution u(x , t) of heat �ow on M � [0,∞), thereexist constants C

0,C

00,C

000and 1

2 < λ < 23 , such that for any δ > 1,

f (x , t) = ln u (x , t) satis�es the following gradient estimate

∑ijXi f j2 � δft +∑

α

(1+ jYαf j2)λ � C0

t+ C

00+ C

000t

λλ�1

with fYαg = f[Xi ,Xj ]g.


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CR Li-Yau gradient estimate

By choosing a frame fT,Z1,Z1g of TM C with respect to the Leviform and f X1, X2g such that

J(Z1) = iZ1 and J(Z1) = �iZ1

andZ1 =

12(X1 � iX2) and Z1 =

12(X1 + iX2),

it follows that

[X1,X2] = �2T and ∆b =12(X 21 + X

22 ) =

12L.

Note that

W (Z ,Z ) = Wx1x 1 and Tor(Z ,Z ) = 2Re (iA11x1x 1)

for all Z = x1Z1 2 T1,0.Shu-Cheng Chang (National Taiwan University)CR Li-Yau Gradient Estimate and Perelman Entropy Formulae

The 10th Paci�c Rim Geometry Conference Osaka-Fukuoka, Part I, Dec. 1-5, 2011 18/ 52

The CR-pluriharmonic Operator

De�nition(Graham-Lee, 1988) Let (M2n+1, J, θ) be a complete pseudohermitianmanifold. De�ne

Pϕ =n

∑α=1(ϕα

αβ + inAβα ϕα)θβ = (Pβ ϕ)θβ, β = 1, 2, � � �, n

which is an operator that characterizes CR-pluriharmonic functions.Here

Pβ ϕ =n

∑α=1(ϕα

αβ + inAβα ϕα)


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Theorem(Chang-Tie-Wu, 2009) Let (M, J, θ) be a closed pseudohermitian3-manifold with nonnegative Tanaka-Webster curvature and vanishingtorsion. If u(x , t) is the positive solution of CR heat �ow on M � [0,∞)such that u is the CR-pluriharmonic function

Pu = 0

at t = 0. Thenjrb f j2 + 3ft �

9t

on M � [0,∞).


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CR Li-Yau Gradient Estimate

Theorem(Chang-Kuo-Lai, 2011) Let (M, J, θ) be a closed pseudohermitian(2n+ 1)-manifold. Suppose that

2Ric (X ,X )� (n� 2)Tor (X , X ) � 0

for all X 2 T1,0 � T0,1. If u (x , t) is the positive solution of�∆b �

∂

∂t

�u (x , t) = 0

with [∆b , T] u = 0 on M � [0, ∞) . Then f (x , t) = ln u (x , t) satis�esthe following subgradient estimate�

jrb f j2 � (1+3n)ft +

n3t(f0)2

�<( 9n + 6+ n)

t.


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subgradient estimate of the logarithm of the positive solution to heat�ow :

TheoremLet (M, J, θ) be a closed pseudohermitian 3-manifold with nonnegativeTanaka-Webster curvature and vanishing torsion. If u(x , t) is the positivesolution of CR heat �ow on M � [0,∞) such that u is theCR-pluriharmonic function

Pu = 0

at t = 0. Then there exists a constant C1 such that u satis�es thesubgradient estimate

jrbuj2

u2� C1

t

on M � (0,∞).


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CR Li-Yau Gradient Estimate for Witten Sublaplacian

We consider the heat equation

∂u (x , t)∂t

= Lu(x , t)

in a closed pseudohermitian (2n+ 1)-manifold (M, J, θ, dµ) with

Lu (x , t) := ∆bu (x , t)�rbφ(x) � rbu (x , t) .

Here dµ = e�φ(x )θ ^ (dθ)n.


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Bakry-Emery pseudohermitian Ricci curvature

The ∞-dimensional Bakry-Emery pseudohermitian Ricci curvature

Ric(L)(W ,W ) := RαβWαWβ + Re[φαβWαWβ]

The m-dimensional Bakry-Emery pseudohermitian Ricci curvature

Ricm,n(L) := Ric(L)� rbφrbφ

2(m� 2n) , m > 2n

Tor(L)(W ,W ) := 2Re[n∑

α,β=1(i(n� 2)Aαβ � φαβ)WαWβ].


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CR Li-Yau Gradient Estimate for Witten Sublaplacian

TheoremLet (M, J, θ) be a closed pseudohermitian (2n+ 1)-manifold. Supposethat

2Ricm,n(L) (X , X )� Tor(L) (X , X ) � 0for all X 2 T1,0 � T0,1. If u (x , t) is the positive solution of�L� ∂

∂t

�u (x , t) = 0 with

[L, T] u = 0

on M � [0, ∞) . Then f (x , t) = ln u (x , t) satis�es the following Li-Yautype subgradient estimate�

jrb f j2 � (1+3n)ft +

n3t(f0)2

�<

m2nt

�9n+ 6+ n

�.


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CR Harnack inequality

TheoremLet (M, J, θ) be a closed pseudohermitian (2n+ 1)-manifold. Supposethat

2Ricm,n(L) (X , X )� Tor(L) (X , X ) � 0for all X 2 T1,0 � T0,1. If u (x , t) is the positive solution of�L� ∂

∂t

�u (x , t) = 0 with

[L, T] u = 0

on M � [0, ∞) . Then for any x1, x2 in M and 0 < t1 < t2 < ∞, we havethe Harnack inequality

u(x2, t2)u(x1, t1)

��t2t1

��[m( 9n +6+n)2n(1+ 3n )

]

expf� (1+3n )

4[dcc (x1, x2)2

(t2 � t1)]g.


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CR Li-Yau Gradient Estimate

Note that

[L, T] u = 2 ImQu � 4Re(φαuβAαβ) + hrbφ0,rbui

and[∆b , T] u = 2 ImQu.

Here Q is the purely holomorphic second-order operator de�ned by

Qu = 2i(Aαβuα)β.


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CR Li-Yau-Hamilton Inequality

Theorem

(Chang-Kuo, 2011) Let (M, J, θ) be a closed spherical pseudohermitian3-manifold with positive Tanaka-Webster curvature and vanishing torsion.Then

4jrbW j2

W 2 � Wt

W� 1t� 0

under the CR Yamabe �ow

∂

∂tθ (t) = �2W (t) θ (t) , θ (0) = θ.

Furthermore, we get a subgradient estimate of logarithm of the positiveTanaka-Webster curvature

jrbW j2

W 2 � 14t

for all t 2 (0,T ).Shu-Cheng Chang (National Taiwan University)CR Li-Yau Gradient Estimate and Perelman Entropy Formulae

The 10th Paci�c Rim Geometry Conference Osaka-Fukuoka, Part I, Dec. 1-5, 2011 28/ 52

CR Li-Yau-Hamilton Inequality

Theorem

(Chang-Kuo, 2011) Let (M, J, θ) be a closed spherical pseudohermitian3-manifold with positive Tanaka-Webster curvature and vanishing torsion.Then

4jrbuj2

u2� utu� 1t� 0

under the time-dependent CR heat equations with potentials evolving bythe CR Yamabe �ow�

∂∂t θ (t) = �2W (t) θ (t) ,∂u∂t = 4∆bu + 2Wu, u0 (x , 0) = 0.


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Perelman Entropy Formulae

TheoremIts monotonicity property of the Perelman entropy functional together withLi-Yau-Perelman reduced distance imply the no local collapsing theoremunder the Ricci �ow.


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TheoremG. Perelman proved that the F -functional

F (gij , ϕ) =ZM(R + jrϕj2)e�ϕdµ

is nondecreasing under the following coupled Ricci �ow(∂gij∂t = �2Rij ,

∂ϕ∂t = �4ϕ+ jrϕj2 � R,

in a closed Riemannian m-manifold (M, gij ).


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TheoremG. Perelman showed that the W-functional

W(gij , ϕ, τ) =ZM[τ(R + jrϕj2) + ϕ�m](4πτ)�

m2 e�ϕdµ, τ > 0

is nondecreasing as well under the following coupled Ricci �ow8><>:∂gij∂t = �2Rij ,

∂ϕ∂t = �4ϕ+ jrϕj2 � R + m

2τ ,dτdt = �1.


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The Ricci �ow∂gij∂t

= �2Rij

coupled with the conjugate heat equation

∂u∂t= �4u + Ru.

1 For u = e�ϕ.2 For u = (4πτ)�

m2 e�ϕ, τ = T � t.


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CR Li-Yau-Perelman Harnack Estimate

Theorem

Let (M, J, θ) be a closed spherical pseudohermitian 3-manifold withnonnegative Tanaka-Webster curvature and vanishing torsion. Under�

∂∂t θ (t) = �2W (t) θ (t) ,∂u∂t = �4∆bu + 4Wu, u0 (x , 0) = 0,

we have∆b f �

34jrb f j2 +

12W � 1

τ� 0

on M � [0,T ) with u = e�f and τ = T � t.


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CR Perelman Entropy Formulae

We de�ne the CR Perelman F -functional by

F (θ(t), f (t)) = 4RM [(∆b f �

34 jrb f j2 + 1

2W )]e�f dµ

=RM (2W + jrb f j2)e�f dµ.

with the constraint RM e

�f dµ = 1.


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Monotonicity Property of CR Perelman Entropy

We derive the following monotonicity property of CR F -functional.

Theorem

Let (M, J, θ) be a closed spherical pseudohermitian 3-manifold withnonnegative Tanaka-Webster curvature and vanishing torsion. Then

ddtF (θ(t), f (t)) = 8

RM

��rH�2f + W

2 Lθ

��2 udµ+ 2RM W jrb f j2 udµ

� 0

under �∂∂t θ (t) = �2W (t) θ (t) ,∂u∂t = �4∆bu + 4Wu, u0 (x , 0) = 0.


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CR Li-Yau-Perelman Harnack Estimate

Theorem

Let (M, J, θ) be a closed spherical pseudohermitian 3-manifold withnonnegative Tanaka-Webster curvature and vanishing torsion. Under�


we have

∆b f �34jrb f j2 +

12W +

f8τ� 12τ� 0

on M � [0,T ) with u = (4πτ)�2e�f and τ = T � t.


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CR Perelman Entropy Formulae

De�ne the CR Perelman W-functional by

W(θ(t), f (t), τ) = 4RM τ[∆b f � 3

4 jrb f j2 + 12W + f

8τ �12τ ]

e�f(4πτ)2

dµ

=RM [τ(2W + jrb f j2) + f

2 � 2](4πτ)�2e�f dµ,

with the constraint RM (4πτ)�2e�f dµ = 1.


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Monotonicity Property of CR Perelman Entropy

Theorem

Let (M, J, θ) be a closed spherical pseudohermitian 3-manifold withnonnegative Tanaka-Webster curvature and vanishing torsion. Then

ddtW(θ(t), f (t), τ(t))

= 8τRM

��rH�2f + W

2 Lθ � 14τLθ

��2 udµ

+τRM [2W jrb f j2 + jrb f j2

τ ]udµ� 0

uner �∂∂t θ (t) = �2W (t) θ (t) ,∂u∂t = �4∆bu + 4Wu, u0 (x , 0) = 0,

for u = (4πτ)�2e�f and τ = T � t.


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CR Li-Yau-Perelman Reduced Distance

Let p, q be two point in M and γ(τ), τ 2 [0, τ], be a Legendrian curvejoining p and q with γ(0) = p and γ(τ) = q.

Theorem

Let (M, J, θ) be a closed spherical pseudohermitian 3-manifold withpositive Tanaka-Webster curvature and vanishing torsion. Under under�


We have

f (q, τ) � 2pτ

τZ0

pτ(W +

18h γ(τ), γ(τ) iLθ

)dτ.


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CR Li-Yau-Perelman Reduced Distance

For

L(γ) =τZ0

pτ(W +

18h γ(τ), γ(τ) iLθ

)dτ,

one can de�ne the CR Perelman reduced distance by

lcc (q, τ) � infγ

2pτL(γ)

and CR Perelman reduced volume by

Vcc (τ) �ZM

(4πτ)�l0 expf� 2pτL(x , τ)gdµ

where inf f is taken over all Legendrian curves γ(τ) joining p, q andL(x , τ) is the corresponding minimum for L(γ).


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The CR Yamabe Shrinking Soliton

TheoremThere is no nontrivial closed shrinking CR Yamabe soliton on a closedpseudohermitian 3-manifold with positive Tanaka-Webster curvature andvanishing pseudohermitian torsion.

Theorem

If (M, J, θ) is a closed spherical CR 3-manifold with vanishing torsion andpositive CR Yamabe constant, then solutions of the CR (normalized)Yamabe �ow converge smoothly to, up to the CR automorphism, a uniquelimit contact form of constant Webster scalar curvature as t ! ∞.


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The Proofs: The CR Bochner Formulae for Sublaplacian

Theorem(Greenleaf, 1986) Let (M2n+1, J, θ) be a complete pseudohermitianmanifold. For a real smooth function u on (M, J, θ),

12∆b jrbuj2 = j(rH )2uj2+ < rbu,rb∆bu >Lθ

+(2Ric � nTor)((rbu)C, (rbu)C)�2i ∑n

α=1(uαuα0 � uαuα0).


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The Proofs: The CR Bochner Formulae

Theorem(Greenleaf, 1986; Chang-Chiu, 2009) Let (M2n+1, J, θ) be a completepseudohermitian manifold. For a real smooth function u on (M, J, θ),

12∆b jrbuj2 = j(rH )2uj2 + (1+ 2

n ) < rbu,rb∆bu >Lθ

+[2Ric + (n� 4)Tor ]((rbu)C, (rbu)C)� 4n < Pu + Pu, dbu >L�θ .


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The Proofs: The CR Bochner Formulae for WittenSublaplacian

Theorem(Chang-Kuo-Lai, 2011) Let (M, J, θ) be a pseudohermitian(2n+ 1)-manifold. For a (smooth) real function f on M and m > 2n, wehave

12Ljrb f j2 � 2(∑n

α,β=1 jfαβj2 +∑nα,β=1α 6=β

jfαβj2) + 1m jLf j2 +

n2 f20

+[2Ricm,n(L)� Tor(L)](rb f ,rb f )+hrb f ,rbLf i+ 2hJrb f ,rb f0i


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The Proofs

De�neF (x , t, a, c) = t

�jrb f j2 (x) + aft + ctf 20 (x)

�.

Theorem

Let�M3, J, θ

�be a pseudohermitian 3-manifold. Suppose that

(2W + Tor) (Z , Z ) � �2k jZ j2

for all Z 2 T1,0, where k is an nonnegative constant. If u (x , t) is thepositive solution on M � [0,∞). Then�

∆b � ∂∂t

�F � 1

a2t F2 � 1

t F � 2 hrb f , rbF i+ t�+�1� c � 2c

a2 F�f 20

+�� 2(a+1)

a2t F � 2k � 2ct

�jrb f j2 + 4ctf0V (f )

i.


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The Proofs

Theorem

Let�M3, J, θ


(2W + Tor) (Z , Z ) � �2k jZ j2

for all Z 2 T1,0, where k is an nonnegative constant. Let a, c , T < ∞ be�xed. For each t 2 [0,T ] , let (p(t), s (t)) 2 M � [0, t] be the maximalpoint of F on M � [0, t]. Then at (p(t), s (t)) , we have

0 � 1a2t F

�F � a2

�+ t

h4 jf11j2 +

�1� c � 2c

a2 F�f 20

+�� 2(a+1)

a2t F � 2k � 2ct

�jrb f j2 + 4ctf0V (f )

i.


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The Proofs

De�ne

V (ϕ) = (A11ϕ1)1 + (A11ϕ1)1 + A11ϕ1ϕ1 + A11ϕ1ϕ1.

Theorem

Let�M3, J, θ


[∆b , T] u = 0.

Then f (x , t) = ln u (x , t) satis�es

V (f ) = 0.


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The Proofs

We claim that for each �xed T < ∞,

F (p(T ), s (T ) , � 4, c) < 163c ,

where we choose a = �4 and 0 < c < 13 . Here

(P(T ), s (T )) 2 M � [0,T ] is the maximal point of F on M � [0,T ].We prove by contradiction. Suppose not, that is

F (p(T ), s (T ) , � 4, c) � 163c .

Due to Proposition ??, (p(t), s (t)) 2 M � [0, t] is the maximal point ofF on M � [0, t] for each t 2 [0,T ]. Since F (p(t), s (t)) is continuous inthe variable t when a, c are �xed and F (p(0), s (0)) = 0, byIntermediate-value theorem there exists a t0 2 (0, T ] such that

F (p(t0), s (t0) , � 4, c) = 163c .


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The Proofs

Hence ��2 (a+ 1)

a2t0F (p(t0), s (t0) , � 4, c)�

2ct0

�= 0

and0 � 1

16s(t0)163c

� 163c � 16

�+�1� c � 2c

16163c

�s (t0) f 20

= 16s(t0)

13c

� 13c � 1

�+� 13 � c

�s (t0) f 20 .

Since 0 < c < 13 , this leads to a contradiction.

HenceF (P(T ), s (T ) , � 4, c) < 16

3c .


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The Proofs

This implies that

max(x , t)2M�[0, T ]

thjrb f j2 (x)� 4ft + ctf 20 (x)

i< 16

3c .

When we �x on the set fTg �M, we have

Thjrb f j2 (x)� 4ft + cTf 20 (x)

i< 16

3c .

Since T is arbitrary, we obtain

jrbuj2

u2� 4ut

u+ ct

u20u2<163ct.

Finally let c ! 13 , then we are done. This completes the proof.


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Thank you very much!


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cr li-yau gradient estimate and perelman entropy …ohnita//2011/slides/part1/shu...cr li-yau...

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